Immersion and clustered coloring

Hadwiger and Hajós conjectured that for every positive integer t, Kt+1-minor free graphs Kt+1-topological minor are properly t-colorable, respectively. Clustered coloring version of these two conjectures which only require monochromatic components to have bounded size has been extensively studied. In this paper we consider the clustered immersion-variant Hadwiger's Hajós' conjecture proposed by Lescure Meyniel independently Abu-Khzam Langston. We determine minimum number required colors H-immersion graphs, any fixed graph H, up a small additive absolute constant. Our result is tight infinitely many H. A key machinery developed in lemma reduces clustering problem on one torsos their tree-cut decomposition or tree-decomposition. byproduct unified proof Alon, Ding, Oporowski Vertigan author Oum about maximum degree minor-closed families.